Applications of Gorenstein projective $\tau$-rigid modules

We first introduce the notion of $CM$-$\tau$-tilting free algebras as the generalization of $CM$-free algebras and show the homological properties of $CM$-$\tau$-tilting free algebras. Then we give a bijection between Gorenstein projective $\tau$-rigid modules and certain modules by using an equivalence established by Kong and Zhang. Finally, we give a partial answer to Tachikawa’s first conjecture by using Gorenstein projective $\tau$-rigid modules.